Labour has failed to bridge the inequality gap and Britain remains a deeply unequal society after almost 13 years of Labour rule, says a government-commissioned report.

The National Equality Panel, set up by Harriet Harman in 2008 to investigate inequalities, found that social class still determines life chances and "deep-seated and systematic differences" remain between social classes.

The report concluded that although earnings inequality had narrowed a little, the large growth of inequality in the 1980s had not been reversed and "social background really matters".

It found that poorer children are less ready to learn when they start school than those from richer families and the gap widens as they progress through the education system. Pupils entering primary school whose mothers had degrees were assessed six months ahead of those who had no qualifications above grade D at GCSE.

Every extra £100 a month in income when children were small was associated with a month's extra development.

Children from low-income homes are also unlikely to go to university and those who did were less likely to attend prestigious institutions or to achieve high class degrees. Meanwhile, within four years of graduation, men who went to private schools earn 8 per cent more than would be expected after allowing for their gender, ethnicity, social class, qualifications and occupation.

The report's findings are politically sensitive following accusations – denied by Gordon Brown – that Labour is embarking on a "class war". They also call Labour's record on tackling inequality into question.